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A second-order recursive sequence.
6

%I #32 Jan 15 2023 02:40:24

%S 1,18,135,660,2475,7722,21021,51480,115830,243100,481338,906984,

%T 1637610,2848860,4796550,7845024,12503007,19468350,29683225,44401500,

%U 65270205,94427190,134617275,189329400,262957500,360988056,490217508,659002960,877549860,1158240600

%N A second-order recursive sequence.

%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

%H Reinhard Zumkeller, <a href="/A056003/b056003.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

%F a(n) = (n+1)*C(n+8, 8).

%F G.f.: (1+8*x)/(1-x)^10.

%F a(n) = A245334(n+8,8)/A000142(8). - _Reinhard Zumkeller_, Aug 31 2014

%F From _Amiram Eldar_, Jan 15 2023: (Start)

%F Sum_{n>=0} 1/a(n) = 4*Pi^2/3 - 266681/22050.

%F Sum_{n>=0} (-1)^n/a(n) = 2*Pi^2/3 - 38656*log(2)/105 + 611409/2450. (End)

%p a:=n->(sum((numbcomp(n,9)), j=9..n)):seq(a(n), n=9..35); # _Zerinvary Lajos_, Aug 26 2008

%t a[n_] := (n+1)*Binomial[n+8, 8]; Array[a, 50, 0] (* _Amiram Eldar_, Jan 15 2023 *)

%o (Haskell)

%o a056003 n = (n + 1) * a007318' (n + 8) 8

%o -- _Reinhard Zumkeller_, Aug 31 2014

%o (PARI) a(n) = (n+1)*binomial(n+8, 8) \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Partial sums of A056117.

%Y Cf. A093644 ((9, 1) Pascal, column m=9).

%Y Cf. A000142, A007318, A052206, A245334, A254142 (partial sums).

%K nonn,easy

%O 0,2

%A _Barry E. Williams_, Jun 12 2000